Directed limits

Abstract definition

A directed limit (or codirected limit) is a limitlim←F\underset{\leftarrow}\lim F of a functor F:J→CF\colon J \to C whose sourcecategoryJJ is a downward-directed set.

More generally, for κ\kappa a regular cardinal say that a κ\kappa-directed setJJ is a poset in which every subset of cardinality <κ\lt \kappa has an upper bound. Then a limit over a functor J→CJ \to C is called κ\kappa-directed limit.

Terminology

Note that the terminology varies. Especially in algebra, a directed limit may be called a ‘projective limit’ or ‘inverse limit’; it's also possible to distinguish these so that an inverse limit may have an arbitrary (possibly undirected) poset as its source. On the other hand, both terms are often used for arbitrary limits as an alternative to the ‘co-’ method of distinction. (The corresponding dual terms are ‘inductive limit’ and ‘direct limit’, with no ‘co-’ even though these are colimits.)

Directed (co)limits were studied in algebra (as projective and inductive limits) before the general notion of limit in category theory. The elementary definition still seen there follows.

Explicit definition

A projective system in CC consists of a directed setII (which we will write directed-upward as usual), a family (Ai)i:I(A_i)_{i: I} of objects of CC, and a family (fij:Aj→Ai)i≤j:I(f_{ij}: A_j \to A_i)_{i \leq j: I} of morphisms, such that: * fii:Ai→Aif_{ii}: A_i \to A_i is the identity morphism on AiA_i; * fik:Ak→Aif_{ik}: A_k \to A_i is the compositefij∘fjkf_{ij} \circ f_{jk}.

Then a projective cone of this projective system is an object XX and a family of projectionsπi:X→Ai\pi_i: X \to A_i such that

πi=fij∘πj. \pi_i = f_{ij} \circ \pi_j .

Finally, a projective limit of the projective system is a projective cone lim←iAi\underset{\leftarrow}\lim_i A_i (where both ff and π\pi are suppressed in the notation, each in its own way) which is universal in that, given any projective cone XX, there exists a unique morphism u:X→lim←iAiu\colon X \to \underset{\leftarrow}\lim_i A_i such that

πi=πi∘u \pi_i = \pi_i \circ u

(where the left-hand π\pi is from the cone XX and the right-hand π\pi is from the limit).

Notice that a projective system in CC consists precisely of a directed set II and a contravariant functor from II (thought of as a category) to CC, while a projective cone or limit of such a projective system is precisely a cone or limit of the corresponding functor. So this is a special case of limit.

As with other limits, a projective limit, if any exists at all, is unique up to a given isomorphism, so we speak of the projective limit of a given projective system.

In algebra

A projective limit in algebra is usually defined as a subalgebra of a cartesian product. To be precise, lim←iAi\underset{\leftarrow}\lim_i A_i consists of those elements (xi)i:I(x_i)_{i: I} of ∏i:IAi\prod_{i: I} A_i such that:

xi=fij(xj). x_i = f_ij(x_j) .

This can be seen as a special case of the construction of an arbitrary limit out of products and equalizers.

Examples

Directed limits over the codirected set (ℕ,≥)(\mathbb{N},\geq) of natural numbers, the tower-diagram,

are extremely common. Classical examples occur in the theory of Postnikov towers and also in the definition of the solenoids.

A ringK[[x]] K [ [ x ] ] of formal power series (for KK a field) is a projective limit of the rings K[x]/xnK[x]/x^n (for nn a natural number). Here, CC is the category of rings, II is the directed set of natural numbers, Ai=K[x]/xiA_i = K[x]/x^i, and fij:Aj→Aif_{ij}: A_j \to A_i is induced by the quotient map K[x]→K[x]/xiK[x] \to K[x]/x^i (which must be proved well defined on K[x]/xjK[x]/x^j for i≤ji \leq j).

Similarly, a ring Zp\mathbf{Z}_p of pp-adic integers (for pp a prime number) is a projective limit of the rings Z/pn\mathbf{Z}/p^n.